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A quantum Kirwan map: Bubbling and Fredholm theory for symplectic vortices over the plane
About this Title
Fabian Ziltener, Korea Institute for Advanced Study, 87 Hoegiro, Dongdaemun-gu, Seoul 130-722, Republic of Korea
Publication: Memoirs of the American Mathematical Society
Publication Year:
2014; Volume 230, Number 1082
ISBNs: 978-0-8218-9472-9 (print); 978-1-4704-1672-0 (online)
DOI: https://doi.org/10.1090/memo/1082
Published electronically: December 20, 2013
Keywords: Symplectic quotient,
Kirwan map,
(equivariant) Gromov-Witten invariants,
(equivariant) quantum cohomology,
morphism of cohomological field theories,
Yang-Mills-Higgs functional,
stable map,
compactification,
weighted Sobolev spaces
MSC: Primary 53D20, 53D45
Table of Contents
Chapters
- 1. Motivation and main results
- 2. Bubbling for vortices over the plane
- 3. Fredholm theory for vortices over the plane
- A. Auxiliary results about vortices, weighted spaces, and other topics
Abstract
Consider a Hamiltonian action of a compact connected Lie group on a symplectic manifold $(M,\omega )$. Conjecturally, under suitable assumptions there exists a morphism of cohomological field theories from the equivariant Gromov-Witten theory of $(M,\omega )$ to the Gromov-Witten theory of the symplectic quotient. The morphism should be a deformation of the Kirwan map. The idea, due to D. A. Salamon, is to define such a deformation by counting gauge equivalence classes of symplectic vortices over the complex plane $\mathbb {C}$.
The present memoir is part of a project whose goal is to make this definition rigorous. Its main results deal with the symplectically aspherical case. The first one states that every sequence of equivalence classes of vortices over the plane has a subsequence that converges to a new type of genus zero stable map, provided that the energies of the vortices are uniformly bounded. Such a stable map consists of equivalence classes of vortices over the plane and holomorphic spheres in the symplectic quotient. The second main result is that the vertical differential of the vortex equations over the plane (at the level of gauge equivalence) is a Fredholm operator of a specified index.
Potentially the quantum Kirwan map can be used to compute the quantum cohomology of symplectic quotients.
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